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Transcript of Burrows Et Al Chapter 9
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introducing inorganic, organic and physical chemistry
second edition
Andrew Burrows
John Holman
Andrew Parsons
Gwen Pilling
Gareth Price
1
Chemistry3
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1Great Clarendon Street, Oxford, OX2 6DP,
United Kingdom
Oxford University Press is a department of the University of Oxford.
It furthers the Universitys objective of excellence in research, scholarship,and education by publishing worldwide. Oxford is a registered trade markofOxford University Press in the UK and in certain other countries
Andrew Burrows, John Holman, Andrew Parsons, Gwen Pilling, GarethPrice 2013
The moralrights of the authors have been asserted
1st Edition copyright 2009
Impression: 1
Allrights reserved. No part of this publication may be reproduced, stored ina retrievalsystem, or transmitted, in any form or by any means, without the
prior permission in writing of Oxford University Press, or as expressly permittedby law, by licence or under terms agreed withthe appropriate reprographics
rights organization. Enquiries concerning reproduction outside the scope of theabove should be sent to the Rights Department, Oxford University Press, at the
address above
Youmust not circulate this workin any other formand youmust impose this same condition on any acquirer
BritishLibrary Cataloguing in Publication Data
Data available
ISBN 9780199691852
Printed and bound in China byC&C Offset Printing Co. Ltd
Links to third party websites are provided by Oxford in good faithandfor information only. Oxford disclaims any responsibility for the materials
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D
This chapter builds on thefollowing topics:
Measurement, units, andnomenclature Section 1.2, p.7
Amount of substance and molarmass Section 1.3, p.18
Balancing equations Section 1.4,
p.21
Concentrations of
solutions Section 1.5, p.34
Enthalpy profiles forexothermic and endothermic
processes Section 1.6, p.45
Chemical equilibrium: how far has
a reaction gone? Section 1.9, p.56
Kinetic molecular theory and thegas laws Section 8.4, p.354
The distribution of molecularspeeds Section 8.5, p.359
Reaction kinetics
yNote that the peaks in the Northern
Hemisphere coincide with the troughsin the Southern Hemisphere.
ZA global plot of methane concentrations in the troposphere showing the variation withtime and latitude. (The colours show the concentration ranges indicated on the vertical
axis.) The concentration is higher in the Northern Hemisphere than it is in the Southern
Hemisphere, because more methane is emitted in the Northern Hemisphere and the
transfer of air across the Equator is slow. NOAA
Concentrationofmethane/p
pm
1.950
Three-dimensional representation of the latitudinal distribution of atmospheric methane in the marine boundary layer. Data from
the Carton Cycle cooperative air sampling network were used. The surface represents data smoothed in time and latitude.
Contact: Dr. Ed Dlugokencky, NOAA ESRL Carbon Cycle, Boulder, Colorado, (303) 4976228, [email protected],
http://wwe.esrl.noaa.gov/gmd/ccgg/.
1.850
1.750
1.650
Latitude
60N30N
30S60S
90S 01 02 0304 05
Year
06 07 08 09 10
0
Increased concentrations of methane in the
troposphere are thought to be due to melting
permafrost in regions of Arctic tundra.z
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Methane in the troposphere
Methane absorbs infrared radiation and is an important
greenhouse gas. To investigate the link between increased
concentrations of greenhouse gases and climate change,
scientists need information about the processes that
release methane into the atmosphere and about what
happens to the methane once there. In particular, they need
to know how quicklythe methane is removed and what is its
average lifetimein the troposphere (the lower atmosphere).
(Box 9.3 on p.396 discusses the significance of the average
lifetime of emitted methane.)
Methane is emitted from a variety of natural and human-
related sources. It is produced when bacteria living in airless
(anaerobic) places break down carbohydrates. The biggest
emissions come from wetlands, ruminant animals (such as
cattle), and biomass burning. The average concentration
of methane in the troposphere in 2010 was 1.79 ppm (by
volume). This is a substantial increase from its pre-industrial
(17501800) concentration of around 0.7 ppm.
The main way methane is removed from the troposphere is
by reaction with hydroxyl radicals, OH. The first step of this
process is
CH4+ OH ; HO + CH
in which the OH radical removes a hydrogen atom from
methane to form a molecule of water and a CHradical.
(You can read about the formation of radicals in Section 19.1,
p.860.) The rate of the reaction of methane with OH is
crucial in determining the lifetime of methane molecules in
the troposphere. The reaction is a simple elementary reaction
(Section 9.4) and the rate of reaction is governed by the
following rate equation
rate of reaction = k[CH4][OH]
where kis the rate constant. The value of the rate constant,
k, is related to how fast the reaction proceeds and is obtained
from laboratory experiments.
In the troposphere, the concentration of methane is
approximately constant (at least on a day to day basisit isincreasing but only slowly), which means methane must be
removed at roughly the same rate as it is being emitted. So,
the average rate of emission = the average rate of removal= k[CH4][OH]
where [CH4] and [OH] are the average concentrations of CH4
and OH, respectively.
Rearranging this equation, the average concentration of
methane in the troposphere is given by
[CH4] =average rate of emission
k[OH]
so that the concentration of methane depends inversely on
the concentration of OH radicals.
The three-dimensional figure opposite shows methane
concentrations in the troposphere over a 10 year period.
There is a seasonal variation because the reactions that
generate OH radicals from water require sunlight, so
concentrations of OH are higher in the summer months.
Methane is, therefore, removed more rapidly in the summer
and its concentration is lower.
Methane concentrations have been rising sharply since 2007
after a decade or so of holding fairly steady. There is concern
that this may be due to the melting of permafraost in regions o
Arctic tundra, which releases previously trapped methane into
the atmosphere.
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Muchof chemistry is about reactions, so a knowledge of the rates at whichthey occur is of
centralimportance. The study of rates of reactions is called reaction kinetics.
Kinetic studies provide information about the mechanisms of reactionsthe detailed
routes by whichthey take place. Most of the reactions youencounter are complex reac-
tions in whichan overallreaction takes place by a series of simple steps, called elementary
reactions. The elementary reactions combine together to give the overallreaction. In thischapter, elementary reactions are discussed first in Section 9.4, followed by complexreactions
in Section 9.5.
9.1 Why study reaction kinetics?
The study of reaction kinetics falls into two major areas. The first is a searchfor fundamental
information about the interactions and energy changes that take place at a molecular level
during an elementary reaction. The progress of a reaction can be charted by constructing
an energy profile suchas the one shown in Figure 9.1. At the highest energy on the curve is
a transient structure, formed from the reactants, that can either return to the reactants, or go
on to form the products. This is the transition state for the reaction. Detailed information
about the structure of transition states contributes to an understanding of how the react-
ants are transformed into the products. Theories of how reactions take place are discussed
in Section 9.8.The second area of reaction kinetics involves the study of complexreactions, and the
elementary reactions that make them up. Experimentalstudies on a reaction in solution
might lead to the determination of a rate equation, whichshows how the rate of a reaction
depends on the concentrations of the reactants, and to a value of the rate constant for the
reaction. Suchexperiments have provided muchof the evidence for the mechanisms of
chemicalreactions, and examples of this are described in Section 9.6.
Rate constants are related to how fast a reaction proceeds at a particular temperature
and compilations of rate constants are widely used in applications of gas phase reaction
kinetics. Rate constants are the key to deciding on the relative importance of reactions in
a system of interconnected reactions, suchas exists in the atmosphere. A knowledge of
rate constants of elementary reactions allowed, for example, the mechanism for ozone
depletion in the stratosphere (upper atmosphere) to be worked out by PaulCrutzen.
In industry, the rates at whichreactions occur are linked to profitability and safety.
Understanding the factors that govern the rates, particularly temperature (Section 9.7),
is essential
. In th
e ph
armaceu
tical
indu
stry, ak
nowl
edge ofk
inetics canh
el
p ch
emistsmaximize the formation of a desired product and minimize competing side-reactions.
An understanding of the role of catalysts in speeding up reactions, including the action
of enzymes in living organisms, comes from kinetic studies. This is discussed in Section 9.9.
First, though, the chapter starts by defining what is meant by the rate of a reaction (Section 9.2)
and looks at ways of monitoring concentrations as a reaction proceeds (Section 9.3).
380 CHAPTER 9 REACTION KINETICS
Potentialenergy
Reactants
Reactants
Transition state
Products
Products
Progress of reaction
Figure 9.1 A general reaction profile
showing the progress of an elementary
reaction from reactants to products, via a
transient, high-energy transition state.
yHelium-filled balloons are sent up into thestratosphere carrying instruments to measure
ozone concentrations, which can be used
to study the kinetics of reactions in the
stratosphere. This balloon is being launched
near McMurdo Station, in Antarctica.
The rates of the reactions involved
in food degradation are important
in determining the shelf life of foods.z
Enthalpy profiles for exothermic and
endothermic reactions are shown in Section 1.6
(p.46). Energy and enthalpy profiles are oftencalled reaction profiles.
Paul Crutzen, Mario Molina, and
F. Sherwood Rowland were awarded
the Nobel Prize in Chemistry in 1995 for their
work concerning the formation and removal of
ozone in the stratosphere. You can read about
some of the reactions involved in t he ozone
layer and the role of CFCs in ozone depletion inBox 27.6 (p.1227).
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9.2 WHAT IS MEANT BY THE RATE OF A REACTION? 381
9.2 What is meant by the rate of a reaction?
The rate of a reaction is the rate at whichreactants are converted into products. Youcan
measure this by monitoring either the consumption of a reactant or the formation of a
product.
Iron powder burninga chemical reaction with a
fast rate.
Iron rustinga chemical reaction with a slow rate.
Consider first a simple reaction in whicha single reactant is converted into a single
product, for example, in an isomerization reaction
reactant ; product
Figure 9.2 plots the concentration of the reactant against time. It shows how the rate of
the reaction changes as the reaction proceeds and is called a kinetic profile. The curve is
steepest at the start, when t= 0, and becomes progressively less steep as the reaction pro-ceeds. At the end of the reaction, the plot becomes horizontalbecause the concentration of
the reactant is no longer changing. The steepness of the curve tells youhow fast the reac-
tion is taking place. The rate of reaction at a particular instant can be found by drawing a
tangent to the curve at that time and measuring the gradient of the tangent. The rate at the
start of the reaction, when t= 0, is called the initial rate of the reaction.
Unlike in a car, which has a
speedometer to measure speed,
reaction rate cannot be measured directly.Instead, you measure the concentration of a
reactant or product at different timesjust
as you could measure the speed of a car
by measuring the distance travelled after
different times.
Gradient gives rateof reaction at time t
[Reactant]
0 t
Rate of reaction
Gradient at t=0 givesinitial rate of reaction
Gradient is zero at theend of the reaction
d[reactant]
dt=
Time
Figure 9.2 The concentration of the reactant plotted against time.
Drawing tangents and
measuring gradientsA tangent is a straight line drawn so that
it just touches the curve at the point of
interest. It has the same slope (gradient)
as the curve at that point.
[Reactant]
c1
c2
t1 t2 Time
Gradient of curve at
point X=gradient of tangentc2 c1t2 t1
=
+ X
In this case, the gradient has a negative value
because the concentration decreases with
time (c2 is smaller than c1). You can find more
information about gradients in Maths Toolkit,
MT4 (p.1312).
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Youcan express the gradient at a particular time as the differential, . The
value of the gradient is negative in Figure 9.2, because the reactant is being used up, so the
rate of reaction = . The minus sign is included because rates are always quoted
as positive quantities.
Now lookat the plot of the concentration of theproductagainst time in Figure 9.3. This
time the gradient of the curve has positive values because the product is being formed in
the reaction. At any time, t, the rate of formation of the product in this reaction is the same
as the rate of consumption of the reactant, so
rate of reaction = =
The relationship between the rate of removalof reactant and the rate of formation ofproduct isnt always this simple and depends on the stoichiometry of the reaction. Take,
for example, the decomposition of hydrogen peroxide to give water and oxygen.
2H2O2 (aq) ; 2H2O(l) + O2 (g)
In this case, 2 molof hydrogen peroxide decompose to form 2molof water and 1molof
oxygen. So, at any time, t, the rate in terms of the formation of oxygen willbe half the rate
in terms of the consumption of H2O2, and half the rate in terms of the formation of H2O,
as shown in Figure 9.4. Another example is given in Worked example 9.1.
In some reactions, the kinetic profile may be a straight line rather than a curve as
shown in Figure 9.5. In this case, the rate of consumption of the reactant is constant and
is given by the gradient of the line. This kind of kinetic profile is explained in Section 9.5
(p.401).
Factors affecting the rate of reaction
The rate of a chemicalreaction may be affected by a number of factors, including:
the concentrations of the reactants;
the temperature of the reaction;
d[reactant]
dt
d[product]
dt
d[reactant]
dt
d[reactant]
dt
382 CHAPTER 9 REACTION KINETICS
The gradient of a plot of [A]
against tis the differential of [A]
with respect to tand is denoted by .
If you need help with differentials,
look in Maths Toolkit, MT6 (p.1318).
d[A]
dt
Gradient givesrate of reaction
at time t
[Product]
0 t
Gradient is zero at theend of the reaction
Gradient at t=0 givesinitial rate of reaction
Time
Rate of reaction = d[product]dt
Figure 9.3 The concentration of the product plotted against time.
The more rigorous definition of
the rate of a reaction on p.386 takes
into account the stoichiometry of the reaction
and ensures that the rate of reaction has
the same value whether given in terms of
consumption of a reactant or the formation
of a product.
The relationship between the amounts
of reactants and products in the balanced
chemical equation is known as the
stoichiometry of the reaction; see
Section 1.4 (p.23).
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the intensity of radiation (if the reaction involves radiation suchas light); the particle size of a solid (related to the surface area of the solid particles);
the polarity of a solvent (for reactions in solution involving ions);
the presence of a catalyst.
The next four sections (Sections 9.39.6) are concerned withthe effects of concentration.
Units of rate of reaction
Concentrations are normally measured in moldm3, so the units for rate of reaction are
usually moldm3 s1.
9.2 WHAT IS MEANT BY THE RATE OF A REACTION? 383
d [H2
O2
]
dt
d [H2
O]
dtRate of reaction = =d [O2]
dt= 2
[H2O2]
Hydrogen peroxide
[O2]
Oxygen
[H2O]
Water
0 Time
Figure 9.4 The decomposition of hydrogen peroxide.
Rate of reaction isconstant and is given
by the gradient ofthe line
[Reactant]
Time0
Figure 9.5 The plot of reactant
concentration against time may
sometimes be a straight line.
For the following reaction, the initial rate of formation ofI2 (aq) was
found to be 2.5 103 moldm3 s1.
S2O82 (aq) + 2I (aq) ; 2 SO42 (aq) +I2 (aq)
What was:
(a) the initial rate of formation of SO42 (aq);
(b) the initial rate of consumption of S2O82 (aq)?
Strategy
Use the stoichiometric equation to find the relationship between
the number of moles ofI2 (aq) formed and the number of moles
of SO42 (aq) formed in the reaction, and the number of moles of
S2O82 (aq) consumed. This gives the relation between the rates of
reaction.
Solution
1 mol of S2O82 (aq) reacts to form 2 mol of SO4
2 (aq) and 1 mol of
I2 (aq).
(a) The initial rate of formation of SO42 (aq)
= 2 initial rate of formationI2 (aq)
= 5.0 103 moldm3 s1.
(b) The initial rate of consumption of S2O82 (aq)
= initial rate of formationI2 (aq)
= 2.5 103 moldm3 s1.
Question
For the reaction between nitrogen and hydrogen
N2 (g) + 3H2 (g) ; 2 NH3 (g)the rate of formation of ammonia was measured as 10 mmol dm3 s1.
What was the rate of consumption of hydrogen?
Worked example 9.1 The rate of a reaction
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9.3 Monitoring the progress of a reaction
The method used to monitor the concentration of a reactant or product as the reaction
proceeds depends very muchon the nature of the reaction, the substances involved, and
the speed withwhichthe reaction takes place.Youneed to lookfor a measurable property of either a reactant or a product, or of the
system as a whole, that changes withtime as the reaction proceeds and from whichthe
concentrations of the reactants can be determined.
Physical properties are often the best because measuring them is non-intrusive and
does not disturb the reaction. For example, youmay have followed a reaction, suchas the
decomposition of hydrogen peroxide or the reaction of marble chips withacid, by collect-
ing and measuring the volume of gas given off at different times during the reaction.
2H2O2 (aq) ; 2H2O(l) + O2 (g)
CaCO3 (s) + 2HCl (aq) ; CaCl2 (aq) + H2O(l) + CO2 (g)
Alternatively, youmay have carried out the reaction in an open conicalflaskon a balance
and measured the change in mass as the reaction proceeded. Figure 9.6 shows some typical
results for these types of experiments.
For reactions taking place in the gas phase (in a closed container of constant volume), it
is often possible to follow the reaction by monitoring the change in pressure as the reaction
proceeds.
Spectroscopic methods are usefulwhere a reactant or product has a strong absorption in
an accessible part of the spectrum. The intensity of absorption can then be used to measure
the concentration of that substance. When the reaction involves a coloured compound,
a spectrophotometer(sometimes called a colorimeter) can be conveniently used to monitor
the concentration of that substance. For example, the reaction of purple [MnO4] ions with
hydrogen peroxide in acidic solution can be monitored in this way
2 [MnO4] (aq) + 5H2O2 (aq) + 6H
+ (aq) ; 2Mn2+ (aq) + 5O2 (g) + 8H2O(l)
Sometimes it is possible to follow a reaction using NMR spectroscopyif there is a char-
acteristic signal that changes during the reaction. For example, in the esterification of
trifluoroethanoic acid (CF3CO2H) withbenzylalcohol(C6H5CH2OH)
CF3CO2H + C6H5CH2OH ; CF3CO2CH2C6H5+ H2O
the consumption of benzylalcoholcan be monitored by the change in the signalfrom theCH2 hydrogens in the
1H NMR spectrum.
Spectroscopic techniques based on the use of flow tubes and lasers have allowed faster and
faster reactions to be monitored. These techniques are discussed at the end of Section 9.4
(p.397) and in Box9.5 (p.402).
384 CHAPTER 9 REACTION KINETICS
Mass ofreactionmixture
and flask
(b)
Time
Volumeof
CO2
(a)
Time
Figure 9.6 Monitoring the reaction of marble chips with dilute hydrochloric acid by (a) measuring the
volume of CO2 given off, and (b) measuring the mass of the reaction mixture and flask as the reaction
proceeds.
Spectroscopic techniques, including
spectrophotometric analysis, are described in
Section 11.4 (p.540). The theory underlying the
techniques can be found in Chapter 10. NMR
spectroscopy is discussed in Sections 10.7
(p.492) and 12.3 (p.576).
Marble is a form of calcium
carbonate, CaCO3.
The relationship between gas pressures,
gas volumes, and the amount in moles is
described in Section 8.1 (p.341).
Lasers that produce a flash ofvisible or ultraviolet radiation on a
femtosecond timescale (1 fs = 1 1015 s)have allowed chemists to probe the
mechanisms of very fast reactions.
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If there is a change in the number or type of ions present in the solution, their con-
centration can be monitored by measuring the conductivityof the solution. For example,
the hydrolysis of 2-chloro-2-methylpropane generates ionic products
(CH3)3CCl (aq) + H2O(l) ; (CH3)3COH(aq) + H+ (aq) + Cl (aq)
Reactions in whichthe concentration of H+ (aq) ions changes can be measured by usinga glass electrode to follow thepH of the solution during the reaction. This would provide
another way of following this hydrolysis reaction.
For reactions of chiralcompounds, where there is a change in optical activity, a polarimeter
can be used to measure the rotation of the plane of polarized light as a function of time. An
early example of the use of this method, first carried out in 1850, was to follow the rate of
the acid-catalysed conversion of sucrose into a mixture of glucose and fructose
Since D-()-fructosehas a larger specific rotation than D-(+)-glucose, the resulting mixture islaevorotatory (rotates plane-polarized light in an anticlockwise direction).
Titrations are sometimes used to measure concentrations in rate studies, but youneed to
withdraw a smallsample to carry out the titration and this disturbs the reaction. It is also a
slow method and there needs to be some way of quenching (stopping or slowing down) the
reaction as soon as youwithdraw the sample.
The key to success in allthe methods is to ensure that there is rapid mixing at the start
(compared to the time the reaction takes) and that the temperature is kept constant. The
rates of many reactions are dependent on temperature so, if the temperature varies, your
results willbe meaningless. This is particularly important for exothermic or endothermic
reactions where there are heat changes during the reaction. There must be good thermal
contact between the reaction vesseland the surroundings (for example, a water bath) to
keep the temperature constant.
9.4 Elementary reactions
Rate equations
Elementary reactions are single-step reactions that involve one or two molecules or
atoms. For example
H
+ Cl2 ; HCl+ Cl
(9.1)
There are two important features of elementary reactions. The first is that the equation
represents the actualchanges that take place at a molecular levelduring the reactionin
the case above, this involves the collision of an H atom witha Cl2 molecule to produce a
molecule of HCland a Clatom. For a complexreaction, whichis made up of a series of
elementary reactions, the chemicalequation merely summarizes the overallstoichiometry
of the reaction. It doesnt tellyouabout the changes that take place at a molecular level.
The second feature of an elementary reaction is that youcan use the chemicalequation
to write a rate equation for the reaction. A rate equation expresses the rate of the reaction
at a particular instant in terms of concentrations of the reactants at that instant.
For the reaction in Equation 9.1, the rate equation is
rate of reaction = k[H][Cl2]
where k is the rate constant for the reaction. The value of k is characteristic of the reaction
and is constant for a particular temperature. The rate constant is related to how fast the
reaction proceeds at that temperature. Figure 9.7 shows plots of the concentration ofreactant against time for different temperatures, that is, for different values of k.
For the elementary reaction
C2H6 ; 2CH3 (9.2)
C H O (aq) H O(l) C H12 22 11-(+)-sucrose
2 6D
+ ; 112 6-(+)-glucose
6 12 6-( -f
O (aq) C H O (aq)D D )
+ rructose
9.4 ELEMENTARY REACTIONS 385
Measurement of the conductivity ofionic solutions is discussed in Section 16.2
(p.726). The use of a glass electrode to
measure pH is described in Section 11.2
(p.520). The definition of pH and calculations
involving pH are in Section 7.2 (p.306).
You can find out about chiral compounds
and the use of a polarimeter to measure optica
activity in Section 18.4 (p.832). The D/L
nomenclature for sugars is explained
in Box 18.6 (p.836).
In Equation 9.1, the hydrogen and
chlorine atoms, H and Cl, are both
radicals.
H+
A radical is an atom or molecule
containing an unpaired electron; see
Section 5.1 (p.217) and Section 19.1 (p.860).
Fast reactions involving a radical
reactant (and their reverse reactions)
are generally single-step elementary
reactions. A complex reaction consists of a
series of elementary steps (see Section 9.4).
You should always assume that a reaction is
complex unless there is evidence otherwise.
A rate equation is sometimes
called a rate law.
Dependence of the rate of reaction on
temperature is discussed in Section 9.7 (p.421)
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386 CHAPTER 9 REACTION KINETICS
[Rea
ctant]
Time0
Low THighT
k>k>kHigh
temperatureLowtemperature
Figure 9.7 Plots of the concentration of
a reactant against time at three different
temperatures, that is, for different values
of the rate constant, k.
Expressing the rate of consumption of a
reactant or the rate of formation of a product
as a differential is described in Section 9.2
(p.381).
the rate equation is
rate of reaction = k[C2H6]
Defining the rate of a reaction
For the reaction of H with Cl2 in Equation 9.1, the rate of reaction has the same valuewhether youconsider the rate of consumption of a reactant or the rate of formation of a
product. For the reaction of ethane to form two methylradicals in Equation 9.2, this is not
the case. The rate of formation of methyl radicals is twice the rate of consumption of
ethane (see p.382). In the reverse process, shown in Equation 9.3
2 CH3 ; C2H6 (9.3)
the rate of consumption of methylradicals is twice the rate of formation of ethane.
It would be helpfulto have a definition for the rate of a reaction that gives the same
value of the rate whether you are monitoring the rate of consumption of any of the
reactants or the rate of formation of any of the products. This is done as follows. For a
generalreaction
aA + bB ; pP + qQ
rate of reaction = (9.4)
This corresponds to the expressions for the rates given in Section 9.2 (p.381), except that
the terms , , etc. have been added to take account of the stoichiometry of the reaction.
This definition is used to write the differentialrate equations for elementary reactions in
the rest of the chapter.
Reaction order
In general, for an elementary reaction
a A + bB ; products
the rate equation takes the form
rate of reaction = k[A]a [B]b (9.5)
where kis the rate constant and a and b are the stoichiometric coefficients in the chemical
equation. In the rate equation they give the order of the reaction:
a is the order of the reaction withrespect to A;
b is the order of the reaction withrespect to B.
The overall orderof the reaction is given by a+ b.Why do rate equations take this form? If the elementary reaction involves two molecules
A and B, these molecules must collide in order to react. The rate of collision of A and B
depends on how many A and B molecules are present in a given volumein other words,
on their concentrations. There is more about collision theory of reaction in Section 9.8
(p.429).
For Equation 9.1, the reaction is first order withrespect to H and first order withrespect to [Cl2]. Overall, the reaction is second order (a+ b= 2).
For Equation 9.2, the reaction is first order withrespect to [C2H6]. Overall, the reac-
tion is first order. (There is only one reactant, so no substance B in the rate equation.)
1
b
1
a
= = =1 d[A]d d[B]d d[P]d d[Q]da t b t p t q t 1 1 1
The order of a reaction with respect
to a reactant (in an elementary
reaction) gives the power to which the
concentration of that substance is raised
in the rate equation.
The number 1 is assumed, rather
than written, in the rate equation,
as is the case for stoichiometric coefficients
in chemical equations.
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For Equation 9.3, the reaction is second order with respect to [CH3]. Overall, the
reaction is second order.
In practice, elementary reactions involve either one or two reactants, and a and b can only
take the values 1 or 2. So, the overall order of an elementary reaction can only be first order
or second order.
Differential rate equations
A rate equation where the rate of consumption of a reactant, or the rate of formation of a
product, is written as a differential (see p.381) is known as a differential rate equation.
Using the definition for the rate of reaction in Equation 9.4, you can now write differential
rate equations for the reactions in Equations 9.19.3. These are shown in Table 9.1, expressed
in terms of both the consumption of reactants and the formation of products.
For example, for
2 CH3; C2H6 (9.3)
rate of reaction = = k[CH3]2
So
= 2 k[CH3]2
You can practise writing differential rate equations in Worked example 9.2.
Units of k
Rates of reaction are expressed in moldm3 s1 (see p.383). The units of k depend on the
order of the reaction. For a first order reaction
rate = k[A]
so k = rate /[A], and the units ofk are given by
= s1
For a second order reaction
rate = k[A]2
so k = rate/[A]2 and the units of k are given by
= dm3 mol1 s1mol dm s
(mol dm (mol dm
3 1
3 3
) )
mol dm s
mol dm
3 1
3
d[CH ]
d3
t
=
1
2
d[CH ]
d
d[C H ]
d3 2 6
t t
9.4 ELEMENTARY REACTIONS 387
Table 9.1 Rate equations expressed in terms of differentials showing the relationship between the
rate of consumption of reactants and the rate of formation of products
Reaction Rate of consumption Rate of formation
of reactants of products
(9.1) H + Cl2 ; HCl + Cl
= = k[H][Cl2] = = k[H][Cl2]
(9.2) C2H6 ; 2 CH3
= k[C2H6] = 2 k[C2H6]
(9.3) 2 CH3; C2H6 = 2 k[CH3
]2 = k[CH3]2
d C H
d
2 6[ ]
t
d CH
d3[ ]
t
d CH
d3[ ]
t
d C H
d2 6[ ]
t
d Cl
d
[ ]
t
d HCl
d
[ ]
t
d Cl
d2[ ]
t
d H
d
[ ]
t
Equations involving differentials
are called differential equations,
see Maths Toolkit MT6 (p.1318).
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388 CHAPTER 9 REACTION KINETICS
An integrated rate equation gives
the concentration of a reactant as a
function of time.
y = mx + c
ln[A]t = kt + ln[A]0
The following elementary reaction between ozone and oxygen atoms
occurs in the atmosphere.
O3 + O ; 2 O2
(a) Write down an expression for the rate of the reaction.
(b) What is the order of the reaction with respect to (i) O3, and (ii) O?
What is the overall order of the reaction?
(c) Use differential expressions, as in Table 9.1 (p.387), to write rate
equations in terms of the rates of consumption of reactants and the
rate of formation of the product.
(Note that both O2 and O atoms have two unpaired electrons and
are biradicals; see Section 4.10, p.194. The dots are omitted here for
simplicity.)
Strategy
The reaction is an elementary process, so you can use the chemical
equation to write the rate equation. The stoichiometric coefficients in
the chemical equation give the orders of reaction with respect to
each reactant. Use Equation 9.4 and Table 9.1 to he lp you write the
rate equation in terms of differentials.
Solution
(a) Rate of reaction = k[O3][O].
(b) The reaction is first order with respect to O3 and first order with
respect to O. The overall order of the reaction is 1 + 1 = 2.
(c) From Equation 9.4
rate of reaction =
Combining this equation with the rate equation in (a)
rate of consumption of reactants: = k[O3][O]
rate of formation of products: = 2 k[O3][O]
Question
The alkaline hydrolysis of bromomethane is an elementary reaction
CH3Br + OH
; CH3OH + Br
(a) Write a rate equation for the reaction in terms of the differential for
consumption of CH3Br.
(b) What is the overall order of the reaction?
d [ O ]
d
2
t
=
d [ O ]
d
d [O]
d
3
t t
= =
d [O
d
d [O]
d
d [O
d
3 2] ]
t t t
1
2
Worked example 9.2 Writing differential rate equations for elementary reactions
This screencast explains
the difference between
differential and
integrated rate equations
and walks you through
Box 9.1 to demonstrate
how the integrated rate equation for a first
order reaction is derived from the differential
rate equation for the reaction.
Integrated rate equations
Rate equations tell you the rate of reaction at a particular instant during the reaction.
However, you cant measure rates of reaction directlyexperimental data usually consist
of measurements of concentration of a reactant or product at different times. It would
be useful to have an expression that shows how the concentration varies with time. This
information comes from the integrated rate equation for the reaction.
The integrated rate equation is derived mathematically from the differential rate equa-
tion for the reaction. You can see how this is done for a first order elementary reaction in
Box 9.1. The resulting equation, and how it is used, are described below.
A first order elementary reaction
For a first order elementary reaction
A ; products
the integrated rate equation (Equation 9.6a) is an expression linking the concentration
of the reactant A at a par ticular time, [A]t, to the time, t, the concentration of A at the start
of the reaction, [A]0, and the rate constant, k.
ln[A]t= ln[A]0 kt (9.6a)
So, if you know the concentration of A at the start and a value for the rate constant, the
integrated rate equation allows you to work out the concentration of a reactant or product
at any time during the reaction.
Alternatively, if you can measure the concentrations during the reaction, the integrated
rate equation can be used to find a value for the rate constant,k. This is easily done because
Differentiation and integrationWhen you differentiate an expression
for a variable, y, with respect to a second
variable,x, the result (a differential ) tells
you the rate at which ychanges with respect to
x. Integration is the opposite of differentiation.
In this case, you start with an expression
describing the rate of change, that is, thegradient of a plot (a curve or a straight line)
and integrate it to find an expression for the
curve or straight line itself.
dy
dx
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9.4 ELEMENTARY REACTIONS 389
First order reaction:A productsGradient = k
In[A]0
In[A]
Time0
Figure 9.8 For the first order reaction,
A; products, a plot of ln [A] against tis
a straight line. The rate constant kcan be
found from the gradient.
Equation 9.6a corresponds to the equation for a straight line (y= mx + c), so a plot of
ln[A]tagainst tis a straight line with a gradient k (Figure 9.8). The plot also helps you to
determine the order of the reaction. If the experimental results plotted in this way lead
to a straight line, then the reaction is first order.
Equations 9.6b and 9.6c are two useful alternative ways of writing Equation 9.6a (see
Box 9.1). Equation 9.6c is an equation for exponential decay and tells you about theshape of the plot of [A] against time. A common feature of all first order reactions is that
the concentration of the reactant decays exponentially with time as shown in Figure 9.9.
From Equation 9.4 (p.386) the differential rate equation
for the first order elementary reaction
A ; products
is given by
rate of reaction = = k[A]
To solve this differential equation to obtain a relationship between
[A] and t, first separate these two variables so that [A] is on one side
of the equation and tis on the other.
= kdt
Now integrate both sides of the equation from [A] = [A]0 at t= 0, to
[A] = [A]t at time t. (Note. Go to Maths Toolkit MT7 (p.1321) if you
need help with carrying out integration.)
To do this you need to use the standard integral
x
x
x
xx
xx x x
1
2
1
2
2 1
d[ln ] ln ln= =
[ ]
[ ][ ]
[ ]A
Ad A
Ad
00
t
k t
t
=
d[A]
[A]
d[A]
dt
Then
ln[A]t ln[A]0 = k(t 0)
which can be rearranged to give
ln[A]t = ln[A]0kt (9.6a)
Using the expression, ln = lna lnb, Equation 9.6a can be
written as
ln = kt (9.6b)
A third way of writing this equation makes use of the fact that
lny=xcan be written asy= ex, so Equation 9.6b becomes
= ekt
so that
[A]t = [A]0 ekt (9.6c)
Equation 9.6c has the form of an exponential decay, which is an
important feature of first order reactions. (The properties of exponen-
tial functions, ex (also written exp(x)), are described in Maths Toolkit
MT3 (p.1310) and MT4 (p.1312).)
[ ]
[ ]
A
A
t
0
[ ]
[ ]
A
At
0
a
b
ln [ ] [ ][ ][ ]AAA =
00
t k t t
Box 9.1 Deriving the integrated rate equation for a first order reaction
A second order elementary reaction
For a second order elementary reaction
A + A ; products
the integrated rate equation is given by
+ 2kt (9.7b)1 1
0[ ] [ ] t=
Straight line graphs
The equation for a straight line is
y=mx+c
You can find help with plotting and interpreting straight
line graphs in Maths Toolkit MT4 (p.1312).
Natural logarthms
In means logarithm to the base e,
where e = 2.7183.
So, Inx= logexand is known as a natural logarithm.
Natural logarithms are explained in Maths Toolkit
MT3 (p.1310).
y = mx+ c
= 2kt+1
[A]0
1
[A]t
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390 CHAPTER 9 REACTION KINETICS
This follows the general procedure given for deriving the
integrated rate equation for a first order reaction in Box 9.1.
From Equation 9.4 (p.386) the differential rate equation for the
second order elementary reaction
A +A ; products
is given by
rate of reaction = = k[A]2
so
= 2 k[A]2
Separating the variables, the equation can be rewritten as
= 2kdt
To obtain a relationship between [A] and t, integrate both sides of the
equation from [A] = [A]0 at t= 0, to [A] = [A]tat time t.
(Note that 2kis constant, so goes outside the integral.)
[ ]
[ ][ ]
[ ]A
Ad A
Ad
0
20
2
t
k t
t
=
d[A]
A2[ ]
d[A]
dt
1
2
d [A]
dt
To do this you need to use the standard integral
Then
= 2k(t 0)
which can be rearranged to give
= 2k t (9.7a)
Here are two useful alternative forms of Equation 9.7a
+ 2k t (9.7b)
[A]t = (9.7c)
Equation 9.7c allows you to predict the concentration of A at any time
after the start of the reaction.
[ ]A
1 2 [A]0
0
+ k t
1 1
0[A] [A]t=
1 1
0[A] [A]t
+1 1
0[A] [A]t
= 1
[A][A]
[A]
0
2 0
t
k t t[ ]
x
x
x
x
x
x x x1
2
1
2
22
1 1d=
=
= +
1 1 1
1 2 1x x x
Box 9.2 Deriving the integrated rate equation for a second order reaction
You can see how this is derived from the differential rate equation in Box 9.2. In this
case, a plot of against t is a straight line with gradient 2k (Figure 9.10). Equations 9.7a
and 9.7c are alternative ways of writing Equation 9.7b (see Box 9.2).
Figure 9.11 shows plots of [A] against t for a first order and for a second order reaction
1
[ ] t
[A] =[A]0ekt
Exponential decay
[A]
[A]0
Time0
Figure 9.9 For the first order reaction,
A ; products, a plot of [A] against tis
an exponential decay curve.
Second order:A A productsGradient = +2k
1[A]
Time0
1[A]0
Figure 9.10 For a second order reaction,
A + A ; products, a plot of 1/[A] againsttis a straight line. The rate constant kcan
be found from the gradient.
with the same initial rate. Note that, for the second order reaction, [A] approaches zero
more slowly than for the first order reaction.
Worked example 9.3 illustrates the use of integrated rate equations.
[A]
[A]0
Time0
Second order
First order
Figure 9.11 A comparison of decay curves for a
first order and a second order reaction with the
same initial decay rate.
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9.4 ELEMENTARY REACTIONS 391
Write an integrated rate equation for the decomposition of ethane to
give two methyl radicals in Equation 9.2 (p.385). Explain how a valueof the rate constantkcan be found from experimental measurements
of concentration of ethane at different times during the course of the
reaction.
C2H6 ; 2 CH3 (9.2)
Strategy
The reaction is a first order e lementary reaction, so the integrated
rate equations are given by Equations 9.6(ac) in Box 9.1.
Solution
Use Equation 9.2 to write the differential equation.
= k[C2H6]
Use Equation 9.6a on p.388 to write the integrated rate equation.
ln[C2H6]t= ln[C2H6]0kt
This is the most useful form of the integrated rate equation in this
case because it is the equation of a straight line that can be used to
find the rate constant. A plot of ln [C2H6] against tis a straight line,
with gradient equal to k.
d [C H ]
d
2 6
tQuestion
Write an integrated rate equation for reaction of two methyl radicalsto form ethane in Equation 9.3 (p.386). Explain how a value of the rate
constant kcan be found from experimental measurements of con-
centration of the methyl radical at different times during the course of
the reaction.
2 CH3 ; C2H6 (9.3)
Worked example 9.3 Writing integrated rate equations
In [C2H6]
In [C2H6]0
Time0
Gradient = k
This is called the isolation technique.
Its use in the study of the kinetics of
complex reactions is described in Section 9.5
(p.403).
For example, if [B]0= 1.00moldm3
and [A]0= 0.01moldm3, then the
concentration of B at the end of the reaction
when A is all used up is (1.00 0.01)moldm3
= 0.99 moldm3. So, [B] is almost constant.
Pseudo-first order reactions
In elementary reactions where there is more than one reactant, suchas A + B ; products,the concentrations of both A and B change during the reaction and affect the rate. A useful
method used to study the kinetics involves isolating one reactant at a time by having the
other reactant in large excess. Suppose B is in large excess: this means that the concentration
of B is effectively constant throughout the reaction and any changes in the rate are due to
changes in the concentration of A alone.
rate of reaction = k[A][B]
If [A]
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392 CHAPTER 9 REACTION KINETICS
Gradient = k
= k[Cl2]0
In [H]
In [H]0
Time0
Figure 9.12 Plot of ln [H ] against
tfor the pseudo-first order reaction,
H + Cl2 ; HCl + Cl, when Cl2 is in large
excess.
The half life of a reactant is the time
taken for its concentration to fall to
half its initial value.
Since ln = lna lnb,
ln = (ln1 ln2) = (ln2 ln1)
= ln = ln 22
1
1
2
a
b
The half life provides a useful
indication of the rate constant
for a first order reaction. A reaction with
a large rate constant has a short half life.
In the presence of a large excess of B, the reaction has all the appearance of a first order
reaction. The reaction is said to show pseudo-first order kinetics under these conditions
and k is called the effective rate constant for a fixed concentration of B, [B]0. A plot of
ln [A] against t is a straight line with gradient k[B]0.
The reaction of chlorine molecules with hydrogen atoms in Equation 9.1 (p.385) can be
studied in this way.
H + Cl2 ; HCl + Cl (9.1)
Under the reaction conditions, H is present in only very small concentrations, so [H]
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9.4 ELEMENTARY REACTIONS 393
Time
[A]02
ConcentrationofA
[A]0
[A]04
[A]08
t1/2 t1/2 t1/2
For first order reaction:
In each successive period of time,t1/2, the concentration of A falls to
half its value at the start of theperiod.
In 2k
t1/2 =
Figure 9.13 For a first order reaction, A ; products, the half life of A is independent of its initialconcentration.
Figure 9.13 shows three successive half lives, when the concentration of A falls from
[A]0 ; ; ; . After n successivehalf lives, the concentration of A willbe .
In contrast to a first order reaction, the half life of a second order reaction does depend
on the initialconcentration of the reactant. For the second order reaction
A + A ; products
youcan find the half life of A by substituting [A]t= [A]0/2 at t= t1/2 into the integrated rateequation.
Using the integrated rate equation in the form of Equation 9.7a (Box9.2, p.390)
= 2kt (9.7a)
For [A]t = [A]0/2
= 2kt1/2
Rearranging this to give an expression for t1/2
t1/2 = (9.10)
In this case, the half life gets longer as the concentration of the reactant falls.
A constant half life indicates a first order reaction. If youinspect a set of data of con-
centration against time for a reaction and see that the initialconcentration falls to half its
value in a certain time as in Figure 9.13, and that another concentration falls to half its
value in the same time, the reaction is likely to be first order. To be sure, youwould need to
plot a graphof ln [A] against t to show that this gives a straight line.
In Worked example 9.4, youcan use boththe half life method and the integrated rate
equation to find the order of a reaction and a value for the rate constant. The significance
of the lifetimes of molecules in the atmosphere is discussed in Box9.3.
1
2 [A]0k
2 1 1
0 0 0[ ] [ ] [ ] =
1 1
0[ ] [ ] t
[A]
20
n
[A]
8
0[A]
40[A]
20
The decay of a radioactive isotope
follows an exponential curve and is a first
order process. Its half life is a characteristic
property of a radioactive isotope. Examples are
discussed in various parts of the book: 14C in
Box 3.8 (p.159), 3H (tritium) in Box 25.7 (p.1156)40K in Box 27.9(p.1243), and 98Tc in Box 28.1
(p.1251).
In fact, anyfractional life remains
constant during a first order reaction.
So, for example, if you measure the quarter
life, t1/4, the time for the concentration of a
reactant to fall to a quarter of its initial value,
you find it is independent of the initial
concentration and remains constant through
the course of the reaction.
Photochromic sunglasses darken or lighten in
response to light intensity. This is achi eved by
incorporating a light-sensitive dye into the lensesThe fading of the dye in the presence of light is a
first order process. Manufacturers publish half
lives of the different coloured dyes used.
This screencast walks you
through Figure 9.13 and
explains how to find the
half life of a reaction.
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[Cyclopropane]/10
3moldm
3
0
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
010 20 30 40 50
17.5min 17.5min 17.5min
60
Time/min
t1/2 =17.5min
394 CHAPTER 9 REACTION KINETICS
Worked example 9.4 Isomerization of cyclopropane
cyclopropane propene
t/min 0 5.0 10 20 30 40 50 60
[cyclopropane] / 103 moldm3 1.50 1.23 1.01 0.68 0.46 0.31 0.21 0.14
Show that the reaction is first order and find a value for the rate constant,k, at 750 K.
Strategy
This worked example illustrates two ways of tackling the problem.
1 Using half lives. Plot a graph of [cyclopropane] against time and measure successive half
lives. If the reaction is first order, the half life will be independent of the concentration of
cyclopropane. Then use t1/2= (Equation 9.9) to find a value for the rate constant.
2 Using an integrated rate equation. Use the integrated rate equation for a first order reaction:
ln[A]t= ln[A]0kt(Equation 9.6b) and plot a graph of ln[cyclopropane] tagainst t. (Rememberln = loge.) For a first order reaction, the plot is a straight line with gradient k.
Solution
1 Using half lives
Plot a graph of [cyclopropane] against t, as shown below. Measure at least three half lives.
ln 2
k
This is a common way of tabulating
data. The concentration of
cyclopropane at t= 0 is 1.50 103 moldm3.So, [cyclopropane] has been divided by
103 moldm3 to give the numbers in the table.
Remember: quantity = number unit, sonumber = quantity/ unit (see Section 1.2, p.7).
The measured half lives have a constant value showing that the reaction is first order.
To findk, substitute an average value for the half life into Equation 9.9
When cyclopropane is heated to 750 Kin a closed container, it isomerizes to form propene.
The reaction was monitored using infrared spectroscopy and the following data obtained.
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9.4 ELEMENTARY REACTIONS 395
In([cyclopropane]
/10
3moldm
3)
0.8
0.4
0
0.4
0.8
1.2
1.6
2.0
0 10 20 30 40 50 60
Time/ min
Gradient =0.040min1
1.201
30.05min
t1/2 = (9.9)
so k = = 0.040 min1 = 6.6 104 s1 (1 min1 = s1)
2 Using an integrated rate equation
The data give the following plot for ln[cyclopropane] against t.
1
60
0.693
17 5min.
ln 2
k
The plot is a straight line showing that the reaction is first order.
The gradient of the line = 0.040min1, so
k = 0.040 min1 = 6.6 104 s1
which agrees with the value from the half life method.
Generally speaking, it is preferable to use the integrated rate equation and plot a straight-line
graph to determine k, rather than use the half life method, because it is difficult to find half lives
accurately from experimental data.
Question
The reaction of methyl radicals to form ethane was investigated using the flash photolysis
technique described on p.398. The following results were obtained at 295 K.
t/ 104 s 0 3.00 6.00 10.0 15.0 20.0 25.0 30.0
[CH3 ] /108 moldm3 1.50 1.10 0.93 0.74 0.55 0.45 0.41 0.35
Show that the reaction is second order and find a value for the rate constant at 295 K.
See Maths note on p.381 about
finding the gradient of a straight line.
t/min 0 5.0 10 20 30 40 50 60
In ([cyclopropane]/ 103 moldm3) 0.405 0.207 0.001 0.386 0.777 1.171 1.561 1.966
Note that [cyclopropane] is divided by its
units so that you are finding the logarithm
of a number (rather than a quantity with units).
Here the units of [cyclopropane] are 103 moldm3.
You could have plotted the graph using units ofmoldm3 for [cyclopropane] and obtained the same
gradient. This is because, when you find the
gradient, you are subtracting two values of ln
[cyclopropane]:
lna1 lna
2 = ln and, ln 103a
1 ln103a
2 = ln
a1
a2
a1
a2
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396 CHAPTER 9 REACTION KINETICS
The impact of methane on climate change (see p.379)
depends on the amount emitted from the Earths surface, the length oftime it stays in the troposphere, and its ability to absorb infrared radiation.
This box is about calculating the lifetime of methane in the troposphere.
Methane is removed from the troposphere by reaction with OH
radicals. These radicals act as scavengers and are responsible for
removing many of the compounds we emit
CH4+OH ; CH3 + H2O
This is a second order elementary reaction and the rate of reaction =k[CH4][
OH].
The kinetics of reactions in the troposphere are different from those
in laboratory experiments, because the reactions are not taking place
in a closed container. In the troposphere, OH radicals are constantly
being produced and removed. As a result, the concentration of OH
is approximately constant, [OH]constant, and you can treat its reaction
with CH4 as a pseudo-first order reaction (see p.391).
Rate of reaction = k[CH4], wherek = k[OH]constant
The total concentration of methane in the troposphere is also approx-
imately constant, but to work out a lifetime you need to consider the fate
of specific emitted methane molecules. The lifetime,, of methanemolecules in the troposphere is theaverage time between emission
of a molecule of CH4 and its removal by reaction withOH.
By this definition, is equal to the time it takes for an initial con-centration [CH4]0 to fall to [CH4]0/ eapproximately a third of its
initial value, while in the presence of a constant [OH]. (The lifetime is
related to the probability of a molecule still being present after emis-
sion. This falls exponentially with time, which explains the presence of
e in the expression for .) On the decay curve below, compare withthe half life, t1/2, which is the time it takes for the concentration to fall
to half of its initial value.
Using the integrated rate equation for a first order reaction in the
form of Equation 9.6b (Box 9.1, p.389)
ln = kt (9.6b)
When t= , [CH4]t = [CH4]0/e, so
ln = k, so ln = k
Rearranging to give an expression for
=
But lne = 1, andk = k[OH]constant, so
=
In the troposphere, the average concentration of OH radicals is
1 1015moldm3 andk= 3.9 106 dm3 mol1 s1. Substituting thesevalues in the expression above gives a value for of ~8 years.
The long lifetime of 8 years means that methane travels large dis-
tances from where it was emitted and becomes broadly distributed
throughout the troposphere (see p.379). For a given emission rate,
the tropospheric concentration is larger if the lifetime is long.
Now contrast this with another hydrocarbon, isoprene, which is
released into the troposphere in huge quantities from vegetation. It
absorbs infrared radiation, but is not a significant greenhouse gas. Its
emission rate on a global scale is of the same order of magnitude as that
of methane, but it has a short lifetime ( about 1 hour) so it is only foundclose to its emission source and its tropospheric concentration is small.
1
OH constantk[ ]
=
ln1
e ln e
k k
1
e
CH
e CH0[ ]
[ ]4
4 0
[ ]
[ ]
CH
CH4 t
4 0
Box 9.3 Atmospheric lifetime of methane
Time
[CH4]02
[CH4]0
Concentrationofmethane
[CH4]0e
t1/2
isoprene
[2-methylbuta-1,3-diene]
Question
Calculate a value for the half life (in years) of methane under the
conditions in the troposphere.
YTropical rainforests are a source of isoprene.
YDecay curve of emitted methane showing lifetime, , and half life, t1/2(e = 2.7183)
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9.4 ELEMENTARY REACTIONS 397
Kinetic techniques used to study elementary reactions
Elementary reactions often involve atoms and other radicals and occur very rapidly, so that
special techniques are needed to study them. Two of the most widely used are flow techniques
and flash photolysis. Both methods are designed to produce well-mixed reactants on a very
short timescale. The concentration of one of the reactants is then measured after a shorttime interval, often using a spectroscopic technique. A series of experiments is carried out,
varying the time interval in each experiment, to give values of concentration at different
times after the start of the reaction. The conditions are usually chosen so that one reactant
is in large excess so that the reaction follows pseudo-first order kinetics (see p.391).
The discharge flow method
This is commonly used to study the reactions of atoms with molecules, such as the reaction
of chlorine radicals with ozone, an important reaction in the stratosphere.
Cl + O3 ; ClO
+ O2
The chlorine radicals are formed by flowing chlorine gas (Cl2), diluted in N2, though a
microwave discharge, which causes some of the Cl2 molecules to dissociate to form Cl
radicals. The experimental setup is shown in Figure 9.14.
O3 (also diluted in N2) is then introduced through an injector and mixes rapidly with the
Cl/Cl2/N
2mixture already flowing through the apparatus. The whole mixture then flows
down a tube at a constant speed, u. The Cl radicals are monitored by a detector at a dis-
tance x along the tube, which corresponds to the time (t= x/u) since the reactants were
mixed and the reaction started. The experiment is then repeated and the time varied by
changing the position of the injector. In this way, the concentration of Cl radicals can be
measured for a series of reaction times.
The conditions are arranged so that [Cl]
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Flash photolysis
In one example of this technique, a brief intense flashof light from a laser is used to
produce radicals very rapidly by photolysis of a precursor molecule. The radicals generated
react witha second reactant already in the reaction cell. The concentration of the radical
rises rapidly during the flashand then decays as it reacts. Its concentration is monitored in
a series of experiments as a function of time.
For example, the reaction of OH radicals withSO2 molecules has been studied in this
way. (This reaction is important in the formation of vapour trails from jet aircraft.)
OH + SO2 ; HOSO2
In this case, the reaction cellinitially contains a mixture of H2O2 and SO2. A laser operating
at 248nm photolyses the H2O2 molecules to produceOH radicals
H2O2+ hv ; 2OH
The conditions are arranged so that [ OH]
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9.4 ELEMENTARY REACTIONS 399
rotations of chemicalbonds, and their rearrangements, as product molecules are formed
from the reactants. The laser is effectively freezing the molecules in time and taking a
snapshot. This new area of chemistry is called femtochemistry.
Box9.4 describes one of the earliest uses of flashphotolysis and the significance of the
results almost 50 years later.
Femtosecond lasers are being used to investigate
the rapid transfer of solar energy between
molecules in photosynthesis.
The Egyptian born chemist, Ahmed Zewail, was
awarded the Nobel Prize in 1999 for his development
of femtochemistry, the probing of chemical reactions
on the femtosecond (1 fs = 1 1015 s) timescale. His
technique has been called the worlds fastest camera.
Ronald Norrish and George Porter were awarded the
Nobel Prize in 1967 for their invention of flash photolysis in the late
1940s. One of the earliest publications using the new technique
described the photochemical reaction between Cl2 and O2 and
the observation of an absorption spectrum due to the short-livedintermediate, ClO. Porter then used this spectrum to monitor the
concentration of ClO at different times to study the kinetics of its
decay.
The absorption spectrum was obtained by first firing a photolytic
flash lamp (causes photolysis) to produce ClO radicals and then
firing a second flash lamp a measured time delay after the first. Light
from the second flash passed through the reaction cell and then
through a spectrometer. The spectrum was recorded on a photo-
graphic plate. A series of experiments was conducted with different
time delays between the two flashes.
Figure 1 is a photograph from Porters 1952 paper and shows
a series of spectra recorded at different times after the photolytic
flash. Each spectrum contains a series of bands corresponding to
the vibrational structure of the absorption spectrum in the region of
270 nm. The strength of the ClO absorption decreases with time as
the ClO reacts. The concentration of ClO was determined from the
fraction of light absorbed at a specific wavelength (see Section 10.3,
p.454).
By plotting 1/[ClO] against time and obtaining a straight line,
Porter showed that the decay of ClO is a second order process and
obtained the rate constant for the reaction
ClO + ClO ; products
YFigure 1 ClO spectra recorded at different times after the photolytic flash.
Box 9.4 Using flash photolysis to monitor ClO radicals
Femtochemistry allows chemists
to observe the transition state for a
reaction (see Section 9.8, p.430).
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400 CHAPTER 9 REACTION KINETICS
Fifty years later, interest in this reaction was reawakened and
new studies, based essentially on the same technique as that used
by Porter, were conducted using a modern instrument at the Jet
Propulsion Laboratory in California. The impetus for the new investiga-
tion was the observation of the stratospheric ozone hole by JoeFarman and his colleagues of the British Antarctic Survey at Halley
Bay in Antarctica. ClO is involved in a catalytic cycle leading to ozone
depletion (see Boxes 27.6, p.1227 and 27.8, p.1235).
However, the catalytic cycle cannot explain the dramatic reduc-
tion in ozone in the Antarctic spring when the concentration of
oxygen atoms is very low. (In the cycle, ClO radicals react with O
to regenerate Cl radicals.) An alternative mechanism for forming Cl
from ClO was needed and the second order reaction of ClO, first
investigated by Porter, provided a possible route to Cl, via its dimer,
ClOOCl
ClO + ClO ; ClOOClClOOCl +hv ; Cl + ClOO
ClOO ; Cl + O2
Experiments (like the one in Figure 2) were designed to measure theyields of all the possible products from the reaction ClO + ClO. Theresults showed that dimer formation is the major route to the forma-
tion of Cl radicals from ClO in the Antarctic spring.
Question
Write a differential rate equation and an integrated rate equation for
the formation of the ClO dimer from ClO.
Cooling pipes
Photolytic laser
Reaction cell
Mirrors directlight fromlasers into
the cell
YFigure 2 A modern flash photolysis apparatus set up to monitor the decayof radicals. The photolytic laser is on the left. The probe laser (which fires the
second flash) is off to the right. Mirrors direct the light from the two lasers into the
reaction cell. The red light being reflected by the mirrors is from the probe laser.
Summary
l An elementary reaction is a single step reaction involving one or two molecules or atoms.
l For a general reactionaA +b B ; p P + q Q
rate of reaction =
l The rate equation for an elementary reaction can be written directly from the chemical equation for the reaction.
l A rate equation, in which the rate of consumption of a reactant, or the rate of formation of a product, is written as a differ-
ential, is called a differential rate equation.
l An integrated rate equation gives the concentration of a reactant as a function of time.
l The half life of a reactant is the time taken for its concentration to fall to half its initial value.
l Table 9.2 summarizes the equations for the first order, second order, and pseudo-first order elementary reactions that you
have met in Section 9.4.
= = =1 d [A]
d
d[B]
d
d[P]
da t b t p t q
1 1 1 dd[Q]
dt
Table 9.2 A summary of equations for elementary reactions
Type of reaction Order of reaction Differential rate equation Integrated rate equation Half life, t1/2
A ; products First order = k[A] ln [A]t = ln [A]0 kt
A + A ; products Second order = 2k[A]2 + 2kt
A + B ; products Pseudo-first order = k[A] ln[A]t = ln[A]0 k twhere [A]
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9.5 COMPLEX REACTIONS: EXPERIMENTAL METHODS 401
How do I know if a reaction is
an elementary process or a
complex reaction?Fast reactions involving a radical reactant
(and their reverse reactions) are generally
elementary reactions. You should always
assume that a reaction is a complex reaction
unless you read otherwise. Reactions of the
type A + B + C or A + 2 B must be complexbecause the probability of more than two
reactant molecules colliding simultaneously
is very low.
The mechanism of the acid-catalysed
reaction of propanone with halogens is
discussed in Section 23.3 (p.1079).
This screencast explainsthe difference between
elementary and complex
reactions.
9.5 Complex reactions: experimental methods
Most of the reactions youwillmeet are not simple elementary reactions. They are complex
reactions in whichthe mechanism involves a series of elementary reactions. For a complex
reaction, youcannot write down the overallrate equation directly from the stoichiometricequation for the reaction. The rate equation must be determined from experiments to
investigatehow the rate depends on the concentration of eachof the reactants.
For a generalreaction in whichA and B are reactants
aA + bB ; products
the overallrate equation is given by an equation of the form
rate of reaction = k[A]m[B]n (9.11)
where k is the overall rate constant for the reaction and m and n are the orders of the reaction,
withrespect to A and B, respectively. The orders oftenhave values of 0, 1, or 2, but sometimes
can take higher, or even fractional, values. The overall orderof the reaction is (m+ n).Note that the stoichiometric coefficients in the chemical equation, a and b, do not
appear in the overall rate equation. It may sometimes turn out that m and n take the
values of a and b, but you cannot assume this.
Sometimes a substance that does not appear among the reactants in the stoichiometric
equation can appear in the rate equation. This might be a product of the reaction or a
catalyst. If the rate of the overallreaction is found experimentally to be unaffected by
the concentration of a reactant, the reaction is said to be zero orderwithrespect to that
reactant (m= 0) and it does not appear in the overallrate equation.Here are some examples of complexreactions.
Example 1 The reaction of iodide ions (I) with peroxodisulfate ions (S2O82)
The equation for the reaction is
S2O82 (aq) + 2 I (aq) ; 2SO42 (aq) + I2 (aq)
Experiments show that that the rate equation for the reaction is
rate of reaction = k[S2O82 (aq)][I (aq)]
so this reaction is first order withrespect to S2O82, first order withrespect to I, and second
order overall.
Example 2 The reaction of propanone (CH3COCH3) with iodine (I2)
The equation for the reaction is
CH3COCH3 (aq) + I2 (aq) ; CH3COCH2I(aq) + H+ (aq) + I (aq)
The reaction is catalysed by acids.
Experiments show that the rate equation for the reaction is
rate of reaction = k[CH3COCH3 (aq)][H+ (aq)]
so this reaction is first order withrespect to bothCH3COCH3 and H+. Note that the catalyst,
H+, appears in the rate equation even thoughit is not used up in the reaction. In contrast,
I2 does not appear in the rate equation, even thoughit is a reactant. The reaction is zero
order withrespect to I2. The overallorder of the reaction is two.
Example 3 The reaction between hydrogen (H2) and iodine (I2)
The equation for the reaction is
H2 (g) + I2 (g) ; 2HI(g)
Experiments show that the rate equation for the reaction is
rate of reaction = k[H2 (g)][I2 (g)]
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so this reaction is first order with respect to H2 and first order with respect to I2 and second
order overall. (Further experiments show that this is not an elementary reaction, but a
complex reaction involving I atoms.)
Example 4 The reaction between hydrogen (H2) and bromine (Br2)
The equation for the reaction is
H2 (g) + Br2 (g) ; 2HBr(g)
Experiments show that the rate equation for the reaction is
rate of reaction = k[H2 (g)][Br2 (g)]1/2
so this reaction is first order with respect to H2, but has an order of12
with respect to
bromine. The overall order is 32.
Compare this with Example 3. Similar looking reactions do not always have the same
kinetics.
Choosing an experimental method
To investigate the kinetics of a complex reaction, you need to determine how the rate of
the reaction depends on the concentration of each of the reactantsand also on the con-
centration of any catalyst involved. From these experiments you can find the order of the
reaction with respect to each of these components, then construct an overall rate equation
and find the overall rate constant for the reaction.
The first step is to decide on an experimental method for monitoring the concentration
of one of the reactants (or a product) as a function of time. This is discussed in Section 9.3.
Box 9.5 describes a method for studying fast reactions in solution.
Having decided how you are going to monitor the reaction, you then need to think
about how to carry out the experiments and analyse the results. There is a range of procedures
available. Which one to choose depends on the particular reaction, what information you
require, and the accuracy needed.
402 CHAPTER 9 REACTION KINETICS
The experimentally determined orders
for complex reactions give chemists an insight
into the mechanismsby which reactions occur
(see Section 9.6, p.412). The mechanism of
the reaction H2 +Br2 involves a series of four
elementary reactions and is described in more
detail in Section 9.6 (p.419).
In all kinetics experiments, it is
important to carry out the reactions
at a constant temperature (for example,
in a t hermostatically controlled water bath).
Otherwise, the results will be meaningless.
The stopped-flow technique is used for studying fast
reactions in solution. It is essentially a way of rapidly mixing the
reactants.
Two solutions, containing the separate reactants, are contained
in syringes, which are mechanically driven to force the solutions into
the mixing chamber and then down a short tube and into a third
syringe. The solutions enter the mixing chamber turbulently and this
promotes rapid mixing. After a short time, the plunger in the third
syringe hits a stop and the flow is immediately halted. The concen-
tration of one of the reactants, or of a product, is then monitored
as a function of time using, for example, absorption spectroscopy,
and a kinetic profile is determined.
The kinetics of the reaction (order and rate constant) are determined
by measuring kinetic profiles for a range of initial concentrations.
Typical reaction timescales are ~20 ms.
Diagram of a stopped-flow apparatus.z
Box 9.5 The stopped-flow technique
Drive mechanism
Lightsource
Stop syringe(Syringe 3)
Stop block
Photo-detector
Syringe 1(reactant 1)
Syringe 2(reactant 2)
Mixing chamber
Spectro-photometer
Absorptionmeasurementis made along
this part ofthe flow tube
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9.5 COMPLEX REACTIONS: EXPERIMENTAL METHODS 403
The isolation technique
The most common type of reaction involves more than one reactant. The trouble is that,
as the reaction proceeds, the concentrations of all the reactants decrease and affect the
overallrate of the reaction. This problem is overcome by isolating one reactant at a time,
by having allthe other reactants in large excess. As a result, their concentrations can beassumed to be constant throughout the reaction and the effect of a single reactant on the
rate of reaction can be studied. This is sometimes called the isolation technique.
For a reaction of the type, A + B ; products, where the rate of reaction = k[A]m[B]n,youwould need to conduct two sets of experiments. In the first set, the concentration
of B is in large excess, so that the order with respect to A can be determined. Rate of
reaction = kA [A]m, where k A is an effective rate constant for the reaction of A, and
kA = k[B]n0.
In a second set of experiments, the concentration of A is in large excess, and the order
withrespect to B is determined. Rate of reaction = kB [B]n, where kB is an effective rate con-
stant for the reaction of B, and kB = k[A]0m.
The next taskis to decide what method to use in eachset of experiments to determine
the order withrespect to a single reactant. There are a number of options:
drawing tangents to the concentrationtime curve;
initialrates method;
using integrated rate equations;
using half lives.
Drawing tangents to the concentrationtime curve
First, the readings of concentration of the reactant A against time are used to plot a kinetic
profile. If the plot is a straight line, the rate of reaction is independent of the concentration
of A and the reaction is zero order withrespect to A. In this case, rate of reaction =k and therate constant can be found from the gradient of the straight line.
More likely, the kinetic profile willbe a curve of the type shown in Figure 9.16. Finding
the rate of reaction at different values of [A] involves drawing tangents to the curve at dif-
ferent points and calculating the gradient of the tangent at eachpoint. The gradient of
eachtangent is a measure of the rate of the reaction at that concentration of A.
The isolation technique is introduced
in Section 9.4 (p.391) where pseudo-
first order elementary reactions are discussed.
It is used in conjunction with one of the
methods listed below (except the initial
rates method where it is unnecessary).
This may seem odd. How can the
rate of reaction be independent of one of
the reactants? This is explained on p.413.
Time
ConcentrationofA
0
[A]2
[A]0
[A]1
Gradient of tangent drawn at thispoint gives the rate of the reactionwhen the concentration of A is [A]1
Gradient of tangent drawn at thispoint gives the rate of the reactionwhen the concentration of A is [A]2
Figure 9.16 Drawing tangents to a concentrationtime curve.
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You can then use the values of the rates of reaction determined from the tangents to
plot a graph of the rate against [A]. If this graph is a straight line, the rate of reaction
is directly proportional to [A] and the reaction is pseudo-first order with respect to A.
If the plot is a curve, the reaction is not first order. You cannot, however, assume it
must then be second order with respect to Ait might be third order, or fractional.
You must now plot a graph of the rate against [A]2. If this plot is a straight line, the
reaction is second order with respect to A. If this plot is also a curve, the reaction is
neither first nor second order with respect to A.
A less laborious and more general method of analysing the data involves plotting the
logarithms of the rate and the concentration (called a loglog plot).
rate of reaction = k[A]m
Taking logarithms (either log10 or loge) of each side of the equation gives log (rate) =
log(k[A]m), so that
log(rate) = m log[A] + logk (9.12)
This equation has the form of a straight line, y= mx + c, where m is the gradient and cthe
intercept. A plot of log (rate) against log[A] is always a straight line, whatever the order
of the reaction with respect to A. The order is found from the gradient of the line and the
rate constant is found from the intercept. The sequence of reasoning is summarized inFigure 9.17.
It is difficult to draw accurate tangents to a curveparticularly if there is some scatter
in the experimental data that do not fit on to a smooth curve. To help, there are software
packages that allow you to feed in the concentrationtime data to obtain the gradient as a
404 CHAPTER 9 REACTION KINETICS
The plotis not astraight
line
The reactionis neitherzero, first,nor secondorder with
respect to A
Plot a graphof rate
against [A]2
The reaction issecondorderwith respect
to A
The plot is nota straight line
Draw tangents to thecurve at different points
to find the rate of
reaction for different [A]
Either
Find theorder from
the gradientof the line
The plot is nota straight line
Plot a graphof log(rate)
against log [A]to give a
straight line
Plot a graphof rate
against [A]
The reaction isfirstorder
with respectto A
Measure [A] at differenttimes and plot a graph
of [A] against t
The reaction iszeroorder with
respect to A
The plotis a
straightline
The plotis a
straightline
The plotis a
straightline
Figure 9.17 A summary of the sequence of reasoning to determine the order with respect to A by
drawing tangents to a concentrationtime curve and subsequent graphical methods.
logab = loga+ logb and
logab = b loga
Further help with logarithms can be
found in Maths Toolkit MT3 (p.1310).
y = mx + c
log(rate) = m log[A] + logk
(here m is the order with respect to A)
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9.5 COMPLEX REACTIONS: EXPERIMENTAL METHODS 405
function of concentration. Even so, the method is not used in accurate work. Loglog plots
are quite good for determining reaction orders, but tend to give large errors on the values
of rate constants.
Initial rates methodThe initialrates method provides a more accurate variant of the previous method. As before,
readings of concentration of a reactant (or product) at different times are used to plot a
kinetic profile, but this time only one tangent is drawn to the curve at t= 0. The gradientof the tangent gives the rate of reaction at the start of the reaction. The experiment is then
repeated severaltimes using different initialconcentrations.
For example, to investigate the decomposition of dinitrogen pentoxide
2N2O5 (g) ; 4NO2 (g) + O2 (g)
a series of separate experiments is carried out, eachwitha different initialconcentration of
N2O5. In eachexperiment, the concentration of N2O5 is monitored as the reaction proceeds
and a graphof [N2O5] against time is plotted to give a kinetic profile. For eachgraph, only
the tangent at t= 0 is drawn, so the reaction need only be followed long enoughto allowan accurate tangent to be drawn at this point. The gradients of the tangents give the initial
rate of the reaction for different initialconcentrations of N2O5.
The way to process and interpret the initialrate data to assign a reaction order followsthe reasoning in Figure 9.17. Youcan often get a preliminary idea of the orders by inspect-
ing the tabulated data as described in Worked example 9.5. The most reliable method
is to plot a graph of log(initial rate) versus log(initialconcentration) for each reactant
(see p.404).
Figure 9.18(a) shows the tangents drawn to find the initialrates in five separate experi-
ments. Figure 9.18(b) shows a plot of initialrate against initialconcentration. This is a
straight line, showing that the rate is proportionalto [N2O5] and the reaction is first order.
The initialrate method stillhas the disadvantage of drawing a tangent to a curve, and the
tangent at t= 0 is particularly difficult to draw accurately. This problem can be overcomeby measuring [N2O5], the amount of reactant consumed in a given time t, instead of
drawing the tangent. Provided [N2O5] is less than about 10% of [N2O5]0, is a
good approximation to the initialrate. This is illustrated in Figure 9.19.
[ N O ]2 5t
Time0 0
Experiment 1
Experiment 2
Experiment 3
Experiment 4
Experiment 5
[N2O5]
[N2O5]initial
InitialrateofconsumptionofN2
O5
Experiment 1
Experiment 2
Experiment 3
Experiment 4
Experiment 5
(a) (b)
Figure 9.18 (a) The initial rate of reaction for each experiment (15) is found by drawing a tangent to
the curve at the start of the reaction. (b) A plot of the five initial rates obtained in (a) against the initial
concentration of N2O5 for each experiment.
The initial rate of a reaction is the
instantaneous rate of change in
concentration of a reactant at the instant
the reac